Average Error: 40.5 → 0.6
Time: 7.5m
Precision: 64
Internal Precision: 1344
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0015447488779754344:\\ \;\;\;\;\frac{\frac{e^{x}}{\frac{1}{\sqrt{e^{x} + 1}}}}{\frac{\frac{e^{x + x} \cdot e^{x + x} - 1}{e^{x + x} + 1}}{\sqrt{e^{x} + 1}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{12} + \left(\frac{1}{2} + \frac{1}{x}\right)\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.5
Target40.1
Herbie0.6
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.0015447488779754344

    1. Initial program 0.0

      \[\frac{e^{x}}{e^{x} - 1}\]

    2. Initial simplification0.0

      \[\leadsto \frac{e^{x}}{e^{x} - 1}\]
    3. Using strategy rm

    4. Applied flip--0.0

      \[\leadsto \frac{e^{x}}{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt0.0

      \[\leadsto \frac{e^{x}}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{\color{blue}{\sqrt{e^{x} + 1} \cdot \sqrt{e^{x} + 1}}}}\]

    7. Applied *-un-lft-identity0.0

      \[\leadsto \frac{e^{x}}{\frac{\color{blue}{1 \cdot \left(e^{x} \cdot e^{x} - 1 \cdot 1\right)}}{\sqrt{e^{x} + 1} \cdot \sqrt{e^{x} + 1}}}\]

    8. Applied times-frac0.0

      \[\leadsto \frac{e^{x}}{\color{blue}{\frac{1}{\sqrt{e^{x} + 1}} \cdot \frac{e^{x} \cdot e^{x} - 1 \cdot 1}{\sqrt{e^{x} + 1}}}}\]

    9. Applied associate-/r*0.0

      \[\leadsto \color{blue}{\frac{\frac{e^{x}}{\frac{1}{\sqrt{e^{x} + 1}}}}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{\sqrt{e^{x} + 1}}}}\]

    10. Simplified0.0

      \[\leadsto \frac{\frac{e^{x}}{\frac{1}{\sqrt{e^{x} + 1}}}}{\color{blue}{\frac{e^{x + x} - 1}{\sqrt{1 + e^{x}}}}}\]
    11. Using strategy rm

    12. Applied flip--0.0

      \[\leadsto \frac{\frac{e^{x}}{\frac{1}{\sqrt{e^{x} + 1}}}}{\frac{\color{blue}{\frac{e^{x + x} \cdot e^{x + x} - 1 \cdot 1}{e^{x + x} + 1}}}{\sqrt{1 + e^{x}}}}\]

    if -0.0015447488779754344 < x

    1. Initial program 60.1

      \[\frac{e^{x}}{e^{x} - 1}\]

    2. Initial simplification60.1

      \[\leadsto \frac{e^{x}}{e^{x} - 1}\]

    3. Taylor expanded around 0 0.9

      \[\leadsto \color{blue}{\frac{1}{12} \cdot x + \left(\frac{1}{x} + \frac{1}{2}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0015447488779754344:\\ \;\;\;\;\frac{\frac{e^{x}}{\frac{1}{\sqrt{e^{x} + 1}}}}{\frac{\frac{e^{x + x} \cdot e^{x + x} - 1}{e^{x + x} + 1}}{\sqrt{e^{x} + 1}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{12} + \left(\frac{1}{2} + \frac{1}{x}\right)\\ \end{array}\]

Runtime

Time bar (total: 7.5m)Debug logProfile

herbie shell --seed 2018242 
(FPCore (x)
  :name "expq2 (section 3.11)"

  :herbie-target
  (/ 1 (- 1 (exp (- x))))

  (/ (exp x) (- (exp x) 1)))